Interest is directed to a moving boundary problem with a gradient flow structure which generalizes surface-tension driven Hele-Shaw flow to the case of nonconstant surface tension coefficient taken along with the liquid particles at the boundary. In the case with kinetic undercooling regularization well-posedness of the resulting evolution problem in Sobolev scales is proved, including cases in which the surface tension coefficient degenerates. The problem is reformulated as a vector-valued,

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... enerate parabolic Cauchy problem. To solve this, we prove and apply an abstract result on Galerkin approximations with variable bilinear forms.